# NAG Library Routine Document

## 1Purpose

c06psf computes the discrete Fourier transforms of $m$ sequences, stored as columns of an array, each containing $n$ complex data values.

## 2Specification

Fortran Interface
 Subroutine c06psf ( n, m, x, work,
 Integer, Intent (In) :: n, m Integer, Intent (Inout) :: ifail Complex (Kind=nag_wp), Intent (Inout) :: x(n*m), work(*) Character (1), Intent (In) :: direct
#include nagmk26.h
 void c06psf_ (const char *direct, const Integer *n, const Integer *m, Complex x[], Complex work[], Integer *ifail, const Charlen length_direct)

## 3Description

Given $m$ sequences of $n$ complex data values ${z}_{\mathit{j}}^{\mathit{p}}$, for $\mathit{j}=0,1,\dots ,n-1$ and $\mathit{p}=1,2,\dots ,m$, c06psf simultaneously calculates the (forward or backward) discrete Fourier transforms of all the sequences defined by
 $z^kp=1n ∑j=0 n-1zjp×exp±i2πjkn , k=0,1,…,n-1​ and ​p=1,2,…,m.$
(Note the scale factor $\frac{1}{\sqrt{n}}$ in this definition.) The minus sign is taken in the argument of the exponential within the summation when the forward transform is required, and the plus sign is taken when the backward transform is required.
A call of c06psf with ${\mathbf{direct}}=\text{'F'}$ followed by a call with ${\mathbf{direct}}=\text{'B'}$ will restore the original data.
The routine uses a variant of the fast Fourier transform (FFT) algorithm (see Brigham (1974)) known as the Stockham self-sorting algorithm, which is described in Temperton (1983). Special code is provided for the factors $2$, $3$ and $5$.

## 4References

Brigham E O (1974) The Fast Fourier Transform Prentice–Hall
Temperton C (1983) Self-sorting mixed-radix fast Fourier transforms J. Comput. Phys. 52 1–23

## 5Arguments

1:     $\mathbf{direct}$ – Character(1)Input
On entry: if the forward transform as defined in Section 3 is to be computed, direct must be set equal to 'F'.
If the backward transform is to be computed, direct must be set equal to 'B'.
Constraint: ${\mathbf{direct}}=\text{'F'}$ or $\text{'B'}$.
2:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of complex values in each sequence.
Constraint: ${\mathbf{n}}\ge 1$.
3:     $\mathbf{m}$ – IntegerInput
On entry: $m$, the number of sequences to be transformed.
Constraint: ${\mathbf{m}}\ge 1$.
4:     $\mathbf{x}\left({\mathbf{n}}×{\mathbf{m}}\right)$ – Complex (Kind=nag_wp) arrayInput/Output
On entry: the complex data values ${z}_{\mathit{j}}^{p}$ stored in ${\mathbf{x}}\left(\left(\mathit{p}-1\right)×{\mathbf{n}}+\mathit{j}+1\right)$, for $\mathit{j}=0,1,\dots ,{\mathbf{n}}-1$ and $\mathit{p}=1,2,\dots ,{\mathbf{m}}$.
On exit: is overwritten by the complex transforms.
5:     $\mathbf{work}\left(*\right)$ – Complex (Kind=nag_wp) arrayWorkspace
Note: the dimension of the array work must be at least ${\mathbf{n}}×{\mathbf{m}}+{\mathbf{n}}+15$.
The workspace requirements as documented for c06psf may be an overestimate in some implementations.
On exit: the real part of ${\mathbf{work}}\left(1\right)$ contains the minimum workspace required for the current values of m and n with this implementation.
6:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
 On entry, ${\mathbf{m}}<1$.
${\mathbf{ifail}}=2$
 On entry, ${\mathbf{n}}<1$.
${\mathbf{ifail}}=3$
 On entry, ${\mathbf{direct}}\ne \text{'F'}$ or $\text{'B'}$.
${\mathbf{ifail}}=5$
An unexpected error has occurred in an internal call. Check all subroutine calls and array dimensions. Seek expert help.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

Some indication of accuracy can be obtained by performing a subsequent inverse transform and comparing the results with the original sequence (in exact arithmetic they would be identical).

## 8Parallelism and Performance

c06psf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c06psf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The time taken by c06psf is approximately proportional to $nm\mathrm{log}\left(n\right)$, but also depends on the factors of $n$. c06psf is fastest if the only prime factors of $n$ are $2$, $3$ and $5$, and is particularly slow if $n$ is a large prime, or has large prime factors.

## 10Example

This example reads in sequences of complex data values and prints their discrete Fourier transforms (as computed by c06psf with ${\mathbf{direct}}=\text{'F'}$). Inverse transforms are then calculated using c06psf with ${\mathbf{direct}}=\text{'B'}$ and printed out, showing that the original sequences are restored.

### 10.1Program Text

Program Text (c06psfe.f90)

### 10.2Program Data

Program Data (c06psfe.d)

### 10.3Program Results

Program Results (c06psfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017